2022 AMC 12A Problems/Problem 8
Problem
The infinite product
![\[\sqrt[3]{10} \cdot \sqrt[3]{\sqrt[3]{10}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} \ldots\]](http://latex.artofproblemsolving.com/3/5/1/351d2e676280300007e4b9e7dd3296d6779dd7c5.png)
evaluates to a real number. What is that number?
![$\textbf{(A) }\sqrt{10}\qquad\textbf{(B) }\sqrt[3]{100}\qquad\textbf{(C) }\sqrt[4]{1000}\qquad\textbf{(D) }10\qquad\textbf{(E) }10\sqrt[3]{10}$](http://latex.artofproblemsolving.com/5/8/6/58669832b19a56c4b8fda00b4c2151cba3f003f4.png)
Solution 1
We can write ![$\sqrt[3]{10}$](http://latex.artofproblemsolving.com/9/a/4/9a414a76d3daf242ab82d4069a47d6a9b39428da.png)
as 
. Similarly, ![$\sqrt[3]{\sqrt[3]{10}} = (10 ^ \frac{1}{3}) ^ \frac{1}{3} = 10 ^ \frac{1}{3^2}$](http://latex.artofproblemsolving.com/e/8/7/e8797d24c656db81f5590b6dd1cd96299a42f87c.png)
.
By continuing this, we get the form
which is
.
Using the formula for an infinite geometric series 
, we get
Thus, our answer is 
.
- phuang1024
Solution 2
We can write this infinite product as 
(we know from the answer choices that the product must converge):
![\[L = \sqrt[3]{10} \cdot \sqrt[3]{\sqrt[3]{10}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} \ldots\]](http://latex.artofproblemsolving.com/e/b/d/ebd4f6b0d54dfa39b6afe11fc4f71300ec7d3c6c.png)
If we raise everything to the 
power, we get:
![\[L^3 = 10 \, \cdot \, \sqrt[3]{10} \, \cdot \, \sqrt[3]{\sqrt[3]{10}} \ldots = 10L \implies L^3 - 10L = 0 \implies L \in \{0, \pm \sqrt{10}\}\]](http://latex.artofproblemsolving.com/0/e/e/0eeff502018f3c83c979f22bc580b94a44a298bc.png)
Since is positive (it is an infinite product of positive numbers), it must be that 
.
~ Oxymoronic15
Solution 3
Move the first term inside the second radical. We get
![\[\sqrt[3]{10} \cdot \sqrt[3]{\sqrt[3]{10}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} \ldots = \sqrt[3]{10\sqrt[3]{10}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} \ldots\]](http://latex.artofproblemsolving.com/4/7/2/47245f249f7aa3aa5e0eafff41eead4a9db3bf17.png)
Do this for the third radical as well.
![\[\sqrt[3]{10\sqrt[3]{10}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{10}}} \ldots = \sqrt[3]{10\sqrt[3]{10}\sqrt[3]{\sqrt[3]{10}}} \ldots = \sqrt[3]{10\sqrt[3]{10\sqrt[3]{10\ldots}}}\]](http://latex.artofproblemsolving.com/9/0/4/9042d406a3ef766679a1a442d8fa8f1696abca65.png)
It is clear what the pattern is. Setting the answer as 
we have
![\[P = \sqrt[3]{10P}\]](http://latex.artofproblemsolving.com/9/3/b/93bc6df8c7ece436cd92a22dcc2ffb47bb78bacf.png)
![\[P = \sqrt{10}\]](http://latex.artofproblemsolving.com/9/c/c/9cc977bb66a05a9513a241cfe926eb5d5fce0473.png)
See Also
| 2022 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 7 |
Followed by Problem 9 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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